Derivative of a polynomial
In 4th ESO you already differentiated simple polynomials using the power rule. Now that calculation becomes a working tool: you will differentiate polynomials of any degree quickly and confidently, calculate second and third derivatives, and use them to solve real problems in physics, economics and geometry. The speed and fluency you gain in this section are the foundation of the whole calculus unit.
The power rule and linearity
Calculate the derivative function of the following polynomial function: $f(x) = 5x^4 - 3x^3 + 7x^2 - 2x + 9$.
Calculate the derivative function of the following polynomial function: $f(x) = 5x^4 - 3x^3 + 7x^2 - 2x + 9$.
Calculate the derivative of the function $p(x) = (2x^2 - 3)(x + 4)$ by first expanding the product of the polynomials.
Calculate the derivative of the function $p(x) = (2x^2 - 3)(x + 4)$ by first expanding the product of the polynomials.
The value of the derivative at a point
Once you have obtained the derivative function $f'(x)$, to find the gradient (or rate of change) at a particular point you just substitute its $x$-coordinate into $f'(x)$. There is no need to differentiate again each time.
Find the value of the derivative of the function $g(x) = -2x^3 + 4x^2 - x + 5$ at the point with x-coordinate $x = -1$.
Find the value of the derivative of the function $g(x) = -2x^3 + 4x^2 - x + 5$ at the point with x-coordinate $x = -1$.
The profile of a suspension bridge cable is approximated by the parabola $y = 0,01x^2 - 0,6x + 15$, where $x$ and $y$ are measured in metres. Calculate the gradient of…
The profile of a suspension bridge cable is approximated by the parabola $y = 0,01x^2 - 0,6x + 15$, where $x$ and $y$ are measured in metres. Calculate the gradient of the cable at the point located $40$ metres from the origin.
Stationary points: $f'(x)=0$
Determine the stationary points (those where the derivative is zero) of the quadratic function $h(x) = 3x^2 - 12x + 7$.
Determine the stationary points (those where the derivative is zero) of the quadratic function $h(x) = 3x^2 - 12x + 7$.
Find the coordinates of the points on the graph of the function $f(x) = x^3 - 3x^2 + x - 1$ at which the tangent line is parallel to the angle bisector of the first…
Find the coordinates of the points on the graph of the function $f(x) = x^3 - 3x^2 + x - 1$ at which the tangent line is parallel to the angle bisector of the first quadrant.
Successive derivatives
Calculate the second derivative, $f''(x)$, of the polynomial function $f(x) = x^5 - 4x^4 + 2x^2 - 8$.
Calculate the second derivative, $f''(x)$, of the polynomial function $f(x) = x^5 - 4x^4 + 2x^2 - 8$.
Applications: tangents and optimisation with polynomials
The three previous ideas (differentiating, evaluating at a point, setting equal to zero) solve almost every problem in this unit: gradients of tangents, marginal costs, instantaneous velocities and optimal values.
The production cost, in euros, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 150 + 12x - 0,05x^2$. Find the marginal cost function, $C'(x)$.
The production cost, in euros, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 150 + 12x - 0,05x^2$. Find the marginal cost function, $C'(x)$.
The number of registered users of a productivity application, in thousands, evolves according to the function $U(t) = -t^3 + 9t^2 + 48t + 200$, where $t$ is the time in…
The number of registered users of a productivity application, in thousands, evolves according to the function $U(t) = -t^3 + 9t^2 + 48t + 200$, where $t$ is the time in months since its launch. Determine the growth rate of the number of users at time $t = 5$ months and state whether at that moment the rate of growth is increasing or decreasing.
A company manufactures and sells $x$ units of a tech item. The revenue function is given by $I(x) = 1200x - 2x^2$ and the total cost function is…
A company manufactures and sells $x$ units of a tech item. The revenue function is given by $I(x) = 1200x - 2x^2$ and the total cost function is $C(x) = x^3 - 15x^2 + 400x + 1500$. Determine the number of units that must be produced to maximise profit and calculate this maximum profit.